Positive commutator methods are
also used to prove refined average decay estimates which hold on subsets of the
phase space. Such estimates for the wave and Schrödinger equation
were first derived by Morawetz, using the radial derivative operator
and the generator of the conformal group as multipliers.
These multipliers also work if repulsive interactions are added.
However, modifications are needed if trapped geodesics are present,
and usually only lead to weaker estimates. A major step in this
direction is the use of a sharp localization of the energy, due to
Mourre [1]}. The energy estimate can be obtained by taking
the derivative with respect to time of the expectation value of some
operator, also called propagation observable as in Sigal, Soffer [2]}.
The remarkable paper by Hunziker, Sigal, and Soffer [3]} presents a time-dependent approach
to Mourre theory based on the commutator expansion lemma of Sigal and Soffer. The latter refers to expressing
\([f(A),B]\)  through a series of Taylor type involving higher-order commutators between \(A\)  and \(B\) .
